Distortional Buckling Tests on Cold-Formed Steel Beams...Distortional Buckling Tests on Cold-Formed...

Distortional Buckling Tests on Cold-Formed Steel BeamsCheng Yu1 and Benjamin W. Schafer2

Abstract: Failure in cold-formed steel beams is generally initiated by one of three instabilities: local, distortional, or lateral-torsionalbuckling. For cold-formed steel joists, purlins, or girts, when the compression flange is not restrained by attachment to sheathing orpaneling, distortional buckling may be the predominant failure mode. Experimental results on cold-formed steel beams with unrestrainedcompression flanges are scarce. Therefore a series of distortional buckling tests on cold-formed steel C and Z sections in bending wasconducted to establish the capacity in distortional buckling failures. Test details were selected to allow distortional buckling to form, butrestrict lateral-torsional buckling to the extent possible. These distortional buckling tests also provide a direct comparison against the localbuckling tests previously performed by the writers. As expected, large strength reductions are observed in the tested specimens whendistortional buckling initiated the failure instead of local buckling. U.S., Canadian, and joint North American standards for design, whichare known to primarily focus on local buckling, provided unconservative predictions of the observed strength. The Australian/NewZealand Standard and the direct strength method, which provide explicit methods for calculating the capacity in the distortional bucklingmode, provided reasonably accurate and reliable predictions.

DOI: 10.1061/�ASCE�0733-9445�2006�132:4�515�

CE Database subject headings: Thin wall structures; Steel beams; Buckling; Predictions.

Introduction

Cold-formed steel beams are commonly used in civil constructionas both secondary, e.g., girts and purlins, and primary, e.g., floorjoists, structural members. This paper focuses on two of the mostcommon sections employed in these applications, the C and Zsection. Cold-formed steel C and Z sections are formed from coilsof thin metal �on the order of 1 mm thick� and the resulting crosssection is thin-walled. Thin-walled members must carefullyconsider the role of cross-section instability in their design.

Cross-section instabilities in C and Z section beams include:local buckling, distortional buckling, and lateral-torsional buck-ling. The cross-section deformations associated with each of thethree buckling modes are illustrated in Fig. 1. Local bucklinginvolves distortion of the cross section with only rotation occur-ring at interior fold lines of the section. Distortional bucklinginvolves distortion of the cross section with rotation and transla-tion occurring at interior fold lines. Lateral-torsional buckling ex-cludes distortion of the cross section; however, translation androtation of the entire cross section occur.

The local, distortional, and lateral-torsional buckling modesalso differ greatly in their longitudinal variation along the beam.The longitudinal deformation associated with each of the threebuckling modes is sinusoidal with a half-wavelength as identified

1Assistant Professor, UNT Research Park-Room F115F, Univ. ofNorth Texas, Denton, TX 76207. E-mail: [email protected]

2Assistant Professor, 203 Latrobe Hall, Johns Hopkins Univ.,Baltimore, MD 21218. E-mail: [email protected]

Note. Associate Editor: Donald W. White. Discussion open untilSeptember 1, 2006. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on February 9, 2005;approved on June 27, 2005. This paper is part of the Journal of Struc-tural Engineering, Vol. 132, No. 4, April 1, 2006. ©ASCE, ISSN 0733-
9445/2006/4-515–528/$25.00.
JOUR

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by the minima in Fig. 1. The local buckling mode occurs withrepeated waves at a short length, while lateral-torsional bucklingoccurs in one half-wave over the unbraced length of the beam.Distortional buckling repeats at a wavelength intermediate to thetwo other modes.

The moments associated with each of the three bucklingmodes are given in Fig. 1 as the ratio of the elastic critical buck-ling moment �Mcr� to the moment at first yield �My�. The minimain Fig. 1 provide the critical values. Determination of the bendingstrength, for use in design, requires consideration of these cross-section instabilities, as well as the differing postbuckling charac-teristics in each of the buckling modes, potential interactionamongst the modes, and material yielding.

Fig. 1. Buckling modes of a cold-formed steel Z member inrestrained bending �restrained bending about a horizontal axis,cross-section designated as 8ZS2.25�059, Fy =380 MPa �55 ksi��

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North American specifications for cold-formed steel �CSA1994; AISI 1996; 2001� apply an effective width approach fordetermining the design strength of beams. First an empirical beamstrength curve is used to account for lateral-torsional buckling.Then, to include interaction between lateral-torsional bucklingand the other modes, the gross section modulus is reduced to aneffective section modulus by performing an element by elementreduction via Winter’s effective width formula. This element byelement effective width reduction accounts for local buckling, andin part, distortional buckling.

The effective width reductions for the AISI �1996� and S136�CSA 1994� specifications were based primarily on the experi-mental results of Desmond et al. �1981�. Subsequent experimentson laterally braced C and Z beams indicated that the AISI �1996�and S136 �CSA 1994� specifications do not always provide con-sistent and conservative predictions �e.g., see Hancock et al.�1996�; and Schafer and Peköz �1999��. The two key shortcom-ings identified were lack of a consistent treatment for the interac-tion of elements in local buckling, and inadequate provisions tohandle distortional buckling. Schafer and Trestain �2002� pro-posed a modification, adopted in AISI �2001�, to partiallyaccount for interaction of elements �e.g., the web and flange� inlocal buckling. In addition, testing focused solely on the localbuckling limit state in C and Z sections was completed by Yu andSchafer �2003�.

Distortional buckling remains unaddressed as a separate limitstate in the North American Specification �AISI 2001�. However,based on the work of Hancock �1997� distortional bucklingis considered in Australia �AS/NZS 1996�. Hancock’s methodrequires calculation of the elastic distortional buckling mode,

Table 1. Summary of Specimens Selected for Testing

Performed tests Number

h / t

Min. Max.

Group 1 Z: h , b�d fixed, t varied 7 71.3 138.2

Group 2 Z: h , b�d fixed, t varied 2 126.6 140.4

Group 3 C: h , b d fixed, t varied 8 80.7 241.7

Group 4 C: b , d fixed, h , t varied 7 66.9 186.7

Total 24 66.9 241.7

Note: Min.�minimum and Max.�maximum.

Fig. 2. Definitions for specimen dimensions of C and Z sections

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which may be completed using finite strip analysis as given inFig. 1, or using a closed-form solution �Hancock 1997�. Schaferand Peköz �1999� modified Hancock’s closed-form solution tocover members in which distortional buckling initiates in the web.Silvestre and Camotim �2004� have supplied an alternativeclosed-form solution for elastic distortional buckling based ongeneralized beam theory.

One reason for the lack of an explicit check on distortionalbuckling in AISI �2001� is that definitive experiments do notexist. The original tests of Desmond et al. �1981� used back-to-back specimens which increased the web stiffness. Subsequenttests by researchers �e.g., see the summary in Schafer and Peköz�1999�� included intermittently spaced angles or through-fasteneddecking attached to the compression flange. These testing detailsrestrain the distortional buckling mode. This partially masks anyproblem with AISI �2001� predictions, and also makes existingdistortional buckling design methods �e.g., AS/NZS 1996� appearoverly conservative. However, design situations do exist when thecompression flange is unrestrained. For example, negative bend-ing of continuous members �joists, purlins, etc.� and wind suctionon walls and panels without interior sheathing. Experiments fo-cused on unrestrained distortional buckling failures are needed.

An objective of this work is to determine the experimentalstrength for unrestrained distortional buckling of C and Z sec-tions. Flexural members were chosen for testing, as opposed tocompression members, because distortional buckling is a morelikely mode of failure in standard C and Z sections under flexure.In axial compression of common C and Z sections, local buck-ling, initiated in the web, typically occurs in advance of distor-tional buckling �Schafer 2002�. In flexure, as shown in Fig. 1, thestress gradient in the web increases the local buckling moment,and distortional buckling often controls.

Distortional Buckling Tests

A series of four-point bending tests were conducted on industrystandard C and Z beams. The details of the tests were selected sothat distortional buckling is free to form. The geometry of theselected specimens, testing protocols, and results are explained inthe following sections.

Selection of Specimens

C and Z sections commonly used in the cold-formed steel indus-try were selected for testing. Table 1 summarizes the range of

b / t d / t h /b d /b

. Max. Min. Max. Min. Max. Min. Max.

39.3 7.0 13.4 3.2 3.6 0.28 0.37

42.0 10.1 11.5 3.2 3.3 0.26 0.28

59.1 6.4 20.3 3.8 4.1 0.26 0.35

43.1 6.4 12.9 2.0 6.0 0.19 0.31

59.1 6.4 20.3 2.0 6.0 0.19 0.37

Min

21.9

38.6

20.3

30.9

20.3

specimens considered and the manner in which variation among


Table 2. Geometry and Material Properties for Distortional Buckling Tests

Groupnumber Test label Specimen

h�mm�

bc

�mm�dc

�mm��c

�deg�bt

�mm�dt

�mm��t

�deg�rhc

�mm�rdc

�mm�rht

�mm�rdt

�mm�t

�mm�fy

�MPa�fu

�MPa�

1 D8.5Z120-4E1W D8.5Z120-4 214 67 24 54.2 63 25 50.2 9 9 9 9 3.00 422.7 572.6

D8.5Z120-1 214 67 24 48.1 64 25 52.1 9 9 9 9 3.00 426.4 573.7

D8.5Z115-1E2W D8.5Z115-2 217 65 23 49.0 61 23 48.3 9 9 9 9 2.97 441.9 577.9

D8.5Z115-1 216 68 21 48.3 63 22 48.3 9 9 10 10 2.96 453.3 583.4

D8.5Z092-3E1W D8.5Z092-3 213 66 24 51.9 61 24 51.6 7 7 8 8 2.27 396.8 497.0

D8.5Z092-1 214 66 24 52.4 61 24 50.9 7 7 8 8 2.28 397.9 500.1

D8.5Z082-4E3W D8.5Z082-4 215 64 24 48.5 61 25 51.3 7 7 8 8 2.06 407.9 510.0

D8.5Z082-3 216 64 24 49.9 60 24 49.5 7 7 8 8 2.06 406.4 508.8

D8.5Z065-7E6W D8.5Z065-7 216 63 21 50.0 63 21 49.3 8 8 9 9 1.63 429.6 575.1

D8.5Z065-6 217 63 22 53.0 62 21 48.3 8 8 9 9 1.64 436.4 574.4

D8.5Z065-4E5W D8.5Z065-5 216 60 17 51.3 64 23 47.2 7 7 7 7 1.64 432.6 573.5

D8.5Z065-4 213 61 21 47.3 57 16 51.2 8 8 7 7 1.57 401.4 540.5

D8.5Z059-6E5W D8.5Z059-6 214 62 20 50.4 61 22 48.0 8 8 8 8 1.57 403.4 545.1

D8.5Z059-5 216 61 20 48.3 61 19 48.3 8 8 8 8 1.56 406.8 547.1

2 D11.5Z092-3E4W D11.5Z092-4 285 88 24 48.7 87 23 49.6 8 8 8 8 2.10 481.6 619.5

D11.5Z092-3 286 87 23 49.3 88 22 49.5 8 8 8 8 2.26 483.1 621.8

D11.5Z082-3E4W D11.5Z082-4 290 87 22 48.4 86 22 49.9 8 8 8 8 2.06 507.4 642.2

D11.5Z082-3 288 87 24 50.2 87 24 51.0 8 8 8 8 2.08 494.7 634.0

3 D8C097-7E6W D8C097-7 207 55 16 80.8 54 16 80.0 7 7 7 8 2.54 586.9 625.4

D8C097-6 207 53 16 81.0 53 16 80.0 7 7 7 8 2.55 587.5 632.6

D8C097-5E4W D8C097-5 205 51 17 86.7 51 17 83.0 7 8 7 7 2.54 576.9 625.2

D8C097-4 205 52 17 83.0 51 17 83.0 7 7 7 7 2.53 579.8 627.5

D8C085-2E1W D8C085-2 205 50 16 86.0 50 17 86.6 6 6 6 6 2.10 363.8 453.7

D8C085-1 205 50 16 88.6 50 17 89.0 6 5 6 5 2.15 357.2 442.1

D8C068-6E7W D8C068-6 202 49 17 80.0 50 16 77.8 4 4 4 4 1.80 543.9 557.2

D8C068-7 202 50 16 76.5 50 17 77.5 4 4 4 4 1.80 550.3 556.4

D8C054-7E6W D8C054-7 204 52 14 83.4 52 14 88.7 6 6 5 6 1.34 281.2 361.9

D8C054-6 203 52 15 89.4 52 14 83.3 6 6 6 6 1.32 280.3 350.4

D8C045-1E2W D8C045-1 208 49 17 89.0 49 17 87.6 7 5 6 5 0.88 147.3 294.0

D8C045-2 207 49 17 88.8 49 18 88.3 7 5 6 5 0.88 144.9 293.8

D8C043-4E2W D8C043-4 204 51 13 87.3 51 14 88.8 4 5 4 5 1.17 313.1 420.6

D8C043-2 204 51 13 88.9 50 14 87.7 5 5 5 5 1.20 313.3 420.4

D8C033-1E2W D8C033-2 207 51 17 87.1 49 16 85.8 4 8 5 8 0.86 141.0 289.0

D8C033-1 205 51 16 86.0 50 20 88.0 5 7 5 7 0.86 140.2 290.7

4 D12C068-10E11W D12C068-11 306 52 13 82.0 51 14 85.3 6 6 6 6 1.64 226.7 392.2

D12C068-10 306 51 14 85.9 50 13 94.8 6 6 7 6 1.65 239.1 391.0

D12C068-1E2W D12C068-2 303 52 13 82.5 52 15 77.4 7 6 6 6 1.69 388.0 507.7

D12C068-1 304 54 13 80.6 51 14 83.3 6 6 7 7 1.70 384.9 507.2

D10C068-4E3W D10C068-4 256 51 12 83.2 53 14 83.3 7 5 6 6 1.59 151.7 277.4

D10C068-3 257 53 14 80.7 53 13 81.9 6 6 6 6 1.61 155.3 281.6

D10C056-3E4W D10C056-3 254 50 17 88.0 49 16 89.0 3 4 3 3 1.45 532.4 553.8

D10C056-4 254 49 18 88.6 49 17 87.7 3 4 3 5 1.45 530.0 562.3

D10C048-1E2W D10C048-1 253 52 16 86.1 49 16 79.6 5 5 5 5 1.21 351.9 403.3

D10C048-2 253 51 16 85.7 49 16 83.7 5 5 5 5 1.24 348.8 398.0

D6C063-2E1W D6C063-2 152 51 16 88.7 50 16 87.3 5 4 5 6 1.47 385.4 460.1

D6C063-1 152 51 16 87.0 50 16 86.1 6 4 6 4 1.42 398.4 478.6

D3.62C054-3E4W D3.62C054-4 95 48 10 87.0 48 11 89.0 7 6 7 7 1.41 221.2 369.0

D3.62C054-3 95 48 9 88.0 47 9 88.0 6 7 7 7 1.41 226.8 367.4

Note: Typical specimen label is DxZ�or C�xxx-x. For example, D8.5Z120-1 means the specimen is 216 mm �8.5 in.� high for the web, Z-section, 3.05 mm�0.12 in.� thick and the beam number is 1. Typical test label is DxZ�or C�xxx-xExW. For example, test D8.5Z120-4E1W means the two-paired specimens
are D8.5Z120-4 at the east side and D8.5Z120-1 at the west side.
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the key nondimensional parameters: web slenderness �h / t�, flangeslenderness �b / t�, lip slenderness �d / t�, and web height-to-flangewidth ratio �h /b�, was performed. The dimensions of the speci-mens �Fig. 2� were recorded at midlength and mid-distance be-tween the center and loading points, for a total of three measure-ment locations for each specimen. Mean specimen dimensions areprovided in Table 2.

Testing Setup

The 4.9 m long four-point bending test consists of two C or Zsections in parallel loaded at the 1/3 points, as shown in Figs. 3and 4. The compression flanges of the specimens are unconnectedin the constant moment span, but connected by a through-fastenedpanel �0.48 mm thick, 32 mm high at the rib� in the shear spans.The tension flange of the specimens is connected by an intermit-tently spaced 32�32�1.45 mm angle attached every 305 mmalong the length. Short hot-rolled tube sections �254�191�152�6 mm� stiffen and bolt the two specimens together at theends and load points to minimize shear and web crippling prob-lems. The objective of the testing setup is to allow distortionalbuckling to form in the constant moment span, while restrictinglateral-torsional buckling, and avoiding other limit states.

The loading system employs an 89 kN MTS actuator, whichhas a maximum 152 mm stroke. The test was performed indisplacement control at a rate of 0.0381 mm/s. An MTS 407controller and load cell monitored the force and insured thedesired displacement control was met. Specimen deflections were

Fig. 3. Elevation view of overall test arrangement for distortionalbuckling tests

Fig. 4. Panel setup for distortional buckling tests



Fig. 5. Actuator force-displacement response of distortional bucklingtests—Group 1




Table 3. Distortional Buckling Test Results

Groupnumber

Testlabel Specimen

Mtest

�kN m�My

�kN m�Mcr�

�kN m�Mcrd

�kN m�Mtest /

My

Mtest /MAISI

Mtest /MS136

Mtest /MNAS

Mtest /MAS/NZS

Mtest /MEN1993

Mtest /MDS�

Mtest /MDSd

1 D8.5Z120-4E1W D8.5Z120-4a 28.7 29.9 83.0 44.2 0.96 0.95 0.95 0.95 1.08 1.00 0.96 1.08

D8.5Z120-1 28.7 30.4 83.6 40.9 0.94 0.93 0.93 0.93 1.09 1.00 0.94 1.09

D8.5Z115-1E2W D8.5Z115-2 26.8 30.6 80.4 41.0 0.88 0.86 0.86 0.86 1.02 0.92 0.88 1.02

D8.5Z115-1a 26.8 31.6 78.3 37.5 0.85 0.88 0.88 0.88 1.03 0.93 0.85 1.03

D8.5Z092-3E1W D8.5Z092-3a 17.3 21.0 36.7 23.6 0.82 0.83 0.85 0.82 1.01 0.88 0.82 1.01

D8.5Z092-1 17.3 21.2 37.1 23.7 0.82 0.83 0.85 0.83 1.00 0.87 0.82 1.00

D8.5Z082-4E3W D8.5Z082-4a 14.3 19.9 27.1 18.4 0.72 0.77 0.81 0.76 0.95 0.83 0.77 0.95

D8.5Z082-3 14.3 19.8 27.1 18.7 0.72 0.77 0.81 0.76 0.94 0.83 0.77 0.94

D8.5Z065-7E6W D8.5Z065-7a 10.5 16.5 13.3 10.7 0.64 0.75 0.82 0.75 0.96 0.93 0.81 0.96

D8.5Z065-6 10.5 16.8 13.6 11.6 0.63 0.72 0.79 0.72 0.92 0.88 0.79 0.92

D8.5Z065-4E5W D8.5Z065-5 9.0 16.2 12.3 9.9 0.56 0.70 0.75 0.70 0.86 0.83 0.72 0.86

D8.5Z065-4a 9.0 13.8 12.1 9.4 0.65 0.72 0.80 0.72 0.97 0.90 0.80 0.97

D8.5Z059-6E5W D8.5Z059-6a 8.0 14.6 11.7 9.5 0.55 0.65 0.71 0.65 0.83 0.80 0.70 0.83

D8.5Z059-5 8.0 14.7 11.6 9.4 0.55 0.64 0.70 0.64 0.83 0.79 0.69 0.83

2 D11.5Z092-3E4W D11.5Z092-4 29.6 45.4 34.6 23.8 0.65 0.85 0.92 0.85 1.07 1.15 0.84 1.07

D11.5Z092-3a 29.6 45.7 34.2 23.4 0.65 0.86 0.93 0.86 1.07 1.07 0.84 1.07

D11.5Z082-3E4W D11.5Z082-4a 26.4 44.5 26.0 19.1 0.59 0.86 0.91 0.86 1.06 1.03 0.84 1.06

D11.5Z082-3 26.4 43.7 26.8 20.7 0.60 0.84 0.89 0.84 1.03 1.02 0.84 1.03

3 D8C097-7E6W D8C097-7 23.1 28.4 44.6 32.5 0.81 0.85 0.88 0.85 0.99 0.90 0.83 0.99

D8C097-6a 23.1 28.2 44.3 32.8 0.82 0.85 0.89 0.85 0.99 0.91 0.83 0.99

D8C097-5E4W D8C097-5a 18.7 26.5 42.6 33.5 0.71 0.73 0.76 0.76 0.84 0.77 0.71 0.84

D8C097-4 18.7 26.9 43.4 33.5 0.69 0.72 0.74 0.72 0.83 0.75 0.70 0.83

D8C085-2E1W D8C085-2a 13.8 14.0 24.3 21.6 0.99 0.99 1.02 1.02 1.09 1.03 0.99 1.10

D8C085-1 13.8 14.0 26.2 22.9 0.98 0.98 1.01 1.01 1.07 1.02 0.98 1.07

D8C068-6E7W D8C068-6 11.8 17.8 15.6 15.7 0.67 0.76 0.85 0.83 0.89 0.84 0.82 0.89

D8C068-7a 11.8 18.2 15.8 15.1 0.65 0.77 0.86 0.84 0.89 0.85 0.80 0.89

D8C054-7E6W D8C054-7 5.5 6.9 6.5 7.0 0.79 0.86 0.96 0.85 1.01 0.98 0.95 1.01

D8C054-6a 5.5 6.8 6.3 7.5 0.80 0.86 0.95 0.85 1.00 0.98 0.97 0.99

D8C045-1E2W D8C045-1 1.9 2.5 1.9 3.8 0.75 0.76 0.84 0.83 0.83 0.90 0.96 0.84

D8C045-2a 1.9 2.4 1.9 3.9 0.77 0.77 0.86 0.85 0.85 0.91 0.98 0.84

D8C043-4E2W D8C043-4a 4.8 6.7 4.2 5.3 0.72 0.90 0.97 0.90 1.00 1.03 0.99 1.01

D8C043-2 4.8 6.9 4.6 5.6 0.70 0.86 0.94 0.94 0.98 0.98 0.95 0.97

D8C033-1E2W D8C033-2 1.8 2.3 1.7 3.7 0.82 0.82 0.91 0.90 0.90 0.97 1.05 0.89

D8C033-1a 1.8 2.3 1.7 3.3 0.81 0.82 0.91 0.91 0.93 1.00 1.04 0.92

4 D12C068-10E11W D12C068-11a 10.7 12.1 9.5 10.1 0.88 0.92 1.05 1.05 1.21 1.13 1.12 1.21

D12C068-10 10.7 12.7 9.8 10.7 0.84 0.88 1.00 1.00 1.15 1.07 1.08 1.15

D12C068-1E2W D12C068-2a 11.1 21.2 10.4 11.0 0.52 0.67 0.72 0.70 0.86 0.79 0.79 0.86

D12C068-1 11.1 21.3 10.9 11.4 0.52 0.67 0.72 0.70 0.85 0.78 0.77 0.85

D10C068-4E3W D10C068-4a 5.8 6.0 9.3 9.6 0.95 0.95 1.01 1.01 1.05 1.01 0.98 1.05

D10C068-3 5.8 6.4 10.0 10.8 0.90 0.90 0.94 0.95 0.96 0.94 0.91 0.97

D10C056-3E4W D10C056-3a 9.6 19.6 7.4 10.2 0.49 0.71 0.74 0.74 0.81 0.79 0.80 0.81

D10C056-4 9.6 19.5 7.5 10.9 0.49 0.69 0.72 0.72 0.79 0.76 0.80 0.79

D10C048-1E2W D10C048-1a 7.0 10.8 4.5 6.7 0.65 0.86 0.90 0.90 1.00 1.00 1.03 1.00

D10C048-2 7.0 10.9 4.7 7.0 0.64 0.85 0.89 0.89 0.98 0.97 1.01 0.98

D6C063-2E1W D6C063-2 5.9 7.0 9.8 8.8 0.85 0.94 0.99 0.92 1.00 0.85 0.89 1.00

D6C063-1a 5.9 7.0 8.9 8.0 0.85 0.95 1.01 0.94 1.03 0.85 0.92 1.03

D3.62C054-3E4W D3.62C054-4 1.9 1.9 7.9 4.0 1.04 1.08 1.08 1.08 1.07 1.08 1.04 1.05

D3.62C054-3a 1.9 1.9 7.8 3.4 1.04 1.14 1.14 1.11 1.12 1.10 1.04 1.09a
Controlling specimen, weaker capacity by AISI �1996�.
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measured at the 1/3 points with position transducers. Additionaldetails of the testing setup may be found in Yu and Schafer �2003�and Yu �2005�.

Tension Coupon Tests

Tension tests were carried out following the provisions of ASTME8-00, standard test methods for tension testing of metallicmaterial �ASTM 2000�. Three tensile coupons were taken fromthe end of each specimen: one from the web flat, one from thecompression flange flat, and one from the tension flange flat. Ascrew-driven ATS 900, with a maximum capacity of 44.5 kN wasused for the loading. An MTS 634.11E-54 extensometer was em-ployed to monitor the deformation.

Coupon test results are summarized in Table 2. Greater varia-tion in the yield stress and ultimate-to-yield �fu / fy� ratios wasobserved in the C sections than in the Z sections. Yield stress inthe C sections varied from 140 to 587 MPa, with a range of fu / fy

ratios from 1.01 �for a high yield stress material� to 2.07 �for alow yield stress material�. Yield stress in the Z sections variedfrom 401 to 495 MPa, while the fu / fy ratio was near 1.3 for all


Fig. 9. Comparison of local buckling



specimens. The elastic modulus, E, is assumed to be 203 GPa �seethe tension tests conducted by Yu and Schafer �2003� for furtherdiscussion of this issue�.

Test Results

Table 3 provides a numerical summary of the 24 four-point bend-ing tests that were conducted. Included in Table 3 is the peakmoment observed in each test �Mtest�, the moment to cause firstyield in each specimen based on the tensile coupon results for thatspecimen �My�, and the elastic critical local buckling moment�Mcr�� and distortional buckling moment �Mcrd� as determinedfrom a finite strip analysis �CUFSM, Schafer �2005�, similar toFig. 1�. In addition, Table 3 also provides ratios of test-to-analytically predicted capacities using design methods, including:the American Specification, MAISI �AISI 1996�; the CanadianStandard, MS136 �CSA 1994�; the North American Specification,MNAS �AISI 2001�; the Australia/New Zealand Standard, MAS/NZS

�AS/NZS 1996�; the European Standard EN1993, MEN1993 �CEN2002�; and the direct strength method �DSM�, Schafer and Peköz1998; AISI 2004: Appendix 1—MDS� for local failures, MDSd fordistortional failures�. Figs. 5–8 provide the actuator load-displacement response for the 24 tests. The response is brokeninto the four different geometric groups summarized in Table 1.The test results and interpretations are discussed in detail in thesubsequent sections.

Comparison with Local Buckling Tests

The distortional buckling tests reported in this paper provide anopportunity for comparison with the local buckling tests con-ducted by Yu and Schafer �2003�. The setup for the two seriesof tests is identical, except in the distortional buckling tests thecompression flange is unrestrained in the constant moment span,and in the local buckling tests the compression flange is restrainedby a specially detailed through-fastened deck. Fig. 9 provides theobserved buckling modes in a local and a distortional bucklingtest for nominally identical specimens. In the distortionalbuckling test the buckling wavelength is visibly longer and thecompression flange rotates about the web/compression flangejunction. The observed strength is also significantly reduced.

Among 25 local buckling and 24 distortional bucklingtests, nine pairs of tests use beams with nominally identicalgeometry and material. The observed capacity of these nine tests

stortional buckling tests for 11.5Z092
and di

is provided in Table 4. On average the distortional buckling testsresult in a 17% reduction in the capacity of the section whencompared with the local buckling test.

For a typical Z section �8.5Z092� the actuator load-displacement response and observed failure mechanism in the twotests are provided in Figs. 10 and 11. The load-displacement re-sponse also includes notations for the actuator load which causesfirst yield in the specimen �Py� as well as the actuator load forelastic critical local buckling �PcrL� and distortional buckling�PcrD� as determined from finite strip analysis. For this Z section,and many of the tested specimens, the response is in the inelasticregime, i.e., the elastic local buckling loads �moments� are abovethat to cause first yield. The elastic stiffness is the same in thelocal and distortional buckling tests, but the distortional bucklingtests fails at a lower load and with a slight increase in nonlinear-ity, as shown in Fig. 10. In the distortional buckling test, thedeformations that trigger failure exhibit longer wavelengths in theflange, and little deformation in the web, when compared to thelocal buckling tests.

For a typical C section �8C043� the actuator load-displacementresponse and observed failure mechanism in the two tests areprovided in Figs. 12 and 13. The elastic stiffness in the local anddistortional buckling tests is the same, but failure occurs earlier inthe distortional buckling test. Interestingly, in the local bucklingtest, some postbuckling strength is observed �i.e., the capacity isgreater than the elastic critical local buckling�, but this does notoccur in the distortional buckling test. Failure mechanisms, asshown in Fig. 13, indicate reduced deformations in the web forthe distortional buckling test. The distortional buckling tests on Csections generally exhibit many characteristics consistent withdistortional buckling failures, but the response is more compli-cated than in the Z sections. This issue is discussed in furtherdetail in the subsequent sections.

Failure Modes

Distortional Buckling

The “distortional buckling tests” referred to in this paper allowunrestrained distortional buckling to occur, but they do notguarantee this will happen. Local buckling and even lateral-torsional buckling may still initiate the failure in the distortional

Table 4. Comparison of Nine Pairs of Tests Having the Same Nominal

Pairnumber

Localbuckling test label

Distortionalbuckling test lab

1 8.5Z120-3E2W D8.5Z120-4E1W

2 8.5Z092-4E2W D8.5Z092-3E1W

3 8.5Z082-1E2W D8.5Z082-4E3W

4 8.5Z059-2E1W D8.5Z059-4E3W

5 8C054-1E8W D8C054-7E6W

6 8C043-5E6W D8C043-4E2W

7 12C068-3E4W D12C068-1E2W

8 12C068-9E5W D12C068-10E11W

9 3.62C054-1E2W D3.62C054-3E4W

buckling tests. Of the 24 conducted tests 17 experienced failure

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mechanisms initiated by distortional buckling. A combination ofvisual inspection, recorded deformations, and supplementaryfinite element analyses were used to identify which bucklingmode initiated the observed failure mechanism.

Finite element models consisting of shell elements were devel-oped in ABAQUS �2001� to investigate the influence of the testsetup on the buckling modes. Complete details of the finite ele-ment modeling, including additional nonlinear analysis modelingoutside the scope of this paper, are provided in Yu �2005�. Forspecimen 8C097 Figs. 14–16 present the predicted elastic buck-ling results for lateral-torsional, distortional, and local buckling,respectively. Numerical results for lateral-torsional �McrLTB�, dis-tortional �Mcrd�, and local buckling �Mcr�� from the finite elementmodel are summarized for all the specimens in Table 5. The localand distortional buckling results may be compared with the purecross-section analysis performed by the finite strip method, whichignores the finer details of the test setup, in Table 3. The testingdetails have only a small impact on the elastic buckling moments,on average the increase is less than 5%. The finite element modelsalso indicate the lateral-torsional buckling moment is sufficiently

try and Material Properties

Actuatorpeak load

of local testPL

�kN�

Actuatorpeak load

of distortionaltest PD

�kN�PD/ PL

�%�

77.93 70.59 91

50.40 42.55 84

45.06 35.23 78

27.49 19.75 72

15.53 13.49 87

14.21 11.91 84

38.00 27.40 72

28.94 26.30 91

5.62 4.76 85

Average 83

Fig. 10. Comparison of actuator force-displacement response fortests on 8.5Z092

Geome

el



increased, in all but specimen 8C097, which is discussed furtherbelow.

Based on the elastic buckling results, the Z section beams areanticipated to have failures initiated by distortional buckling,since Mcrd�Mcr� for all specimens. Indeed, visual inspection,recorded deformations, and observed strength all bear this out.

Previous work has shown that even when the elastic criticaldistortional buckling value is higher than the local buckling value�Mcrd�Mcr�� the final failure mechanism may still be triggered bydistortional buckling �Schafer and Peköz 1999; Schafer 2002�.This is due in part to the fact that distortional buckling has lowerpostbuckling reserve, even in the inelastic regime. This findingwas supported by the testing on C sections performed here. Dis-tortional buckling is observed in specimens 8C068, 8C054,8C043, 12C068, and 10C068 as shown in Fig. 17, even thoughMcrd�Mcr�. In general for the C sections, the web/flange juncturein the compression flange does not exhibit rotations as large asthose observed in the Z sections. In the postpeak �past the peakbending moment� range of the tests, the majority of C sectionsexhibit rotation at the web/flange juncture, but in many casestranslation and rotation of the entire section as well. This obser-vation indicates a more complicated collapse response and thepossible interaction of distortional buckling with local/lateral-torsional buckling in the C sections.

In the following sections, an examination of the seven testswhich did not fail in mechanisms initiated by distortional buck-ling are presented. Failure in these seven other tests included: oneby lateral-torsional buckling, one by material yielding, three bylocal buckling, one specimen with large pretest damage, and oneby shear + bending interaction.

Lateral-Torsional Buckling

As shown in Table 5, D8C097-5E4W is the only test specimenwhich has a lower lateral-torsional buckling moment than that oflocal or distortional buckling. In the test, shown in Fig. 18, sig-nificant twist of the specimens occurred. Subsequently, an addi-tional angle was attached to the compression flange at midspanand another test performed �D8C097-7E6W� as shown in Fig. 19.The new test exhibited the same initial elastic stiffness and sus-tained a significantly higher load before failure occurred, asshown in Fig. 20. The large twist that occurred in the first test wasremoved in the subsequent test, as shown in Fig. 21. In the modi-fied test �D8C097-7E6W� failure occurred due to distortional

Fig. 11. Comparison of failur

buckling. D8C097-5E4W, which provides an examination of the



role of lateral-torsional buckling, is not considered a distortionalbuckling test for the purposes of this paper, and is not used incomparison to the design specifications.

Material Yielding

Test D3.62C054-3E4W failed by material yielding. The beam’syield moment is 3.8 kN m and the distortional buckling momentis 7.4 kN m. The observed peak moment was also 3.8 kN m andno deformations consistent with distortional buckling were ob-served prior to obtaining the peak moment. The beam showedsignificant nonlinearity, but no sharp strength loss during theloading process, as shown in Fig. 8.

Local Buckling

Failures initiated by local buckling were visually observed inthree tests: D8C045-1E2W, D8C033-1E2W, and D10C056-3E4W, as shown in Figs. 22–24. Failure in these three tests iscategorized with the previously performed local buckling tests. In

hanisms for tests on 8.5Z092

Fig. 12. Comparison of actuator force-displacement response fortests on 8C043

e mec


addition, according to direct strength method �AISI 2004, Appen-dix 1� predictions these three specimens have lower local buck-ling strength than distortional buckling strength.

Pretest Damage

Test D12C068-1E2W had a dent in the compression flange priorto testing, which initiated the failure during testing. The observedstrength is significantly lower than predictions by design specifi-cations: 23% off the direct strength method �AISI 2004, Appendix1� prediction and 30% off AISI �2001�. This mildly damagedspecimen was examined in greater detail with the use of nonlinearfinite element models in Yu �2005�. For comparison to designspecifications this specimen is not included in the summarystatistics.

Shear + Bending Interaction

Failure in test D8.5Z059-6E5E �the thinnest of the Z memberstested� occurred outside of the constant moment region. Initialgeometric imperfections, uneven specimen setup, and shear andbending interaction are possible reasons for the unexpected fail-ure mode. In all other tests, the tube section bolting the beamstogether provided sufficient restraint to force the failure inside theconstant moment region. For comparison to design specificationsthis specimen is not included in the summary statistics.

Comparison with Design Specifications

Six design methods are considered for comparison: U.S. �AISI1996�; Canada �CSA 1994�; Australia/New Zealand �AS/NZS1996�; North America �AISI 2001�; Europe �CEN 2002�; and therecently adopted direct strength method �Schafer and Peköz 1998;AISI 2004, Appendix 1�. DSM prediction equations for both localand distortional buckling of beams are briefly summarized here.For local buckling the capacity is

�� 0.776, MDS� = Mne �1�

�� 0.776, MDS� = �1 − 0.15�Mcr�

Mne�0.4��Mcr�

Mne�0.4

Mne �2�

�� = �Mne/Mcr� �3�

where Mcr��critical elastic local buckling moment; and

Fig. 13. Comparison of failu

Mne�moment capacity due to lateral-torsional buckling. Since the

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beams are assumed to be laterally braced Mne=My, whereMy�moment at first yield.

For distortional buckling the capacity is

�d � 0.673, MDSd = My �4�

�d � 0.673, MDSd = �1 − 0.22�Mcrd

My�0.5��Mcrd

My�0.5

My �5�

�d = �My/Mcrd �6�

where Mcrd�critical elastic distortional buckling moment.A summary of the test-to-analytically predicted capacities for

all six design methods, for both the local and distortional bucklingseries of tests, is given in Table 6. Results for individual tests areprovided in Table 3. On average, all six methods provide reliablestrength predictions for the local buckling tests. The directstrength method �DSM� uses a single strength curve for eachbuckling mode and gross section properties, while the other fivemethods apply effective width concepts.

For distortional buckling, only AS/NZS, EN1993, and DSMhave specific methodologies. AS/NZS and DSM employ the mini-mum of separate local and distortional buckling strength predic-tions while EN1993 assumes distortional buckling is a reductionto be applied in addition to local buckling. Table 6 shows thatAS/NZS, EN1993, and DSM provide reasonable strength predic-tions for the distortional buckling failures, though EN1993 stillremains about 4% unconservative on average. AS/NZS and DSM

chanisms for tests on 8C043

Fig. 14. Finite element prediction for lateral-torsional buckling modeof beam D8C097-5E4W

re me





Table 5. Elastic Buckling Moments of Performed Tests using ABAQUSModel with all Testing Details

Test labelMcr�

�kN m�Mcrd

�kN m�McrLTB

�kN m�

D8.5Z120-4E1W 171.8 87.2 137.7

D8.5Z115-1E2W �163.9 81.3 132.6

D8.5Z092-3E1W 77.3 50.8 84.4

D8.5Z082-4E3W 56.6 39.8 �63.9

D8.5Z065-7E6W 28.6 23.8 �34.1

D8.5Z065-4E5W 23.3 19.6 �26.0

D8.5Z059-6E5W 24.6 20.7 �28.8

D11.5Z092-3E4W 71.6 52.4 �85.9

D11.5Z082-3E4W 56.5 43.5 �68.7

D8C097-5E4W 89.9 68.9 60.3

D8C085-2E1W 50.9 44.8 �52.9

D8C068-6E7W 33.2 31.6 �38.2

D8C054-7E6W 13.2 14.6 �16.3

D8C045-1E2W 4.1 �4.8 �4.8

D8C043-4E2W 9.7 �12.1 �12.1

D8C033-1E2W 3.7 �4.7 �4.7

D12C068-1E2W 22.8 23.5 �28.9

D12C068-10E11W 20.7 21.4 �26.2

D10C068-4E3W 20.8 21.0 �25.6

D10C056-3E4W 15.8 �18.7 �18.7

D10C048-1E2W 10.0 �10.0 �10.0

D6C063-2E1W 20.4 18.4 �21.5

D3.62C054-3E4W �13.6 7.5 10.6

Note: Lower bounds are given to those modes which are not included inthe first 30 eigenmodes calculated in the ABAQUS analysis. Each modelincludes two specimens, and is thus approximately double the results of

Fig. 15. Finite element prediction for distortional buckling mode ofbeam D8C097-5E4W

Table 3.

Fig. 16. Finite element prediction for local buckling mode of beamD8C097-5E4W

Fig. 17. Observed distortional buckling deformations and failure mechanisms in C sections even when elastic critical local buckling is at a lowermoment than distortional buckling �Mcr��Mcrd�


Fig. 18. Lateral-torsional buckling observed in test D8C097-5E4W

Fig. 19. Distortional buckling observed in re-rest of D8C097-7E6W,note added angle at midspan

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Fig. 20. Actuator force-displacement response for tests on D8C097

Fig. 21. Beam rotation at the south loading point for tests onD8C097

Fig. 22. Local failure observed in test D8C045-1E2W



employ the same basic procedure and are quite similar for distor-tional buckling strength prediction. AS/NZS and DSM provide atest-to-predicted ratio closest to 1 and have the lowest coefficientof variation. AISI, S136, and NAS provide systematically uncon-servative predictions for the distortional buckling strength, withan average error between 8 and 15%.

DSM provides good agreement with the results of both seriesof tests. Fig. 25 provides a graphical representation of the DSMpredictions, in the figure Mcr represents Mcr� for the local buck-ling tests and Mcrd for the distortional buckling tests. Prediction ofthe local buckling capacity exhibits less scatter than the distor-tional buckling capacity. The DSM strength equations were cali-brated to other tests, including a variety of cross sections beyondC and Z sections as summarized in �AISI 2004, Appendix 1�. Thereliable performance of DSM with the test data presented hereserves as additional validation for DSM.

Discussion

Design can account for the worst-case scenario for distortionalbuckling with simple equations such as those used in AS/NZS�1996� or DSM �AISI 2004, Appendix 1� if the elastic distortionalbuckling moment �Mcrd� is known. However, without computermethods such as finite strip analysis, calculation of Mcrd is non-trivial. The writers are working on simplified Mcrd calculationsfor use in preliminary design, but simple, general, and accurateprocedures remain elusive.





One alternative to traditional design methods is to performtesting to determine the distortional buckling strength. As theseexperiments show, generalized protocols for performing the testsare not easy to arrive at. Thicker specimens tend to have bothelevated local and distortional buckling stress and can thus readilybe controlled by lateral-torsional buckling, or yielding. Thinnerspecimens may also be controlled by other limit states, includinglocal buckling and shear and bending interaction. While thesefailure modes were relatively uncommon in our tests, they wereall observed, and serve to demonstrate the variety of behavior thatmay occur even at modest unbraced lengths.

Testing performed herein is a worst-case scenario for distor-tional buckling: �1� there is no restraint of the compression flange,�2� the demand is pure bending, and �3� lateral-torsional bucklingis restricted. Restraint of the compression flange is a key issue, ifrestraint does not exist distortional buckling is likely to initiatethe failure at levels much below that of local buckling. The caseof partial restraint is examined experimentally in Yu and Schafer�2003�. Further testing, analysis, and design procedures areneeded to incorporate this influence in design. The pure bendingdemand used in the testing setup does not allow for an explorationof the influence of moment gradient on the distortional bucklingcapacity. Since the distortional buckling half-wavelength is rela-tively long, moment gradients will have an influence on the dis-tortional buckling capacity. Finally, if a beam is not braced,lateral-torsional buckling will often occur far in advance of dis-tortional buckling, and will control the design. Issues related topartial restraint and moment gradient in distortional buckling ofcold-formed steel beams have been recently examined in Yu�2005� and will be the subject of future papers by the writers.

Conclusions

Four-point bending tests on a wide variety of industry standard,laterally braced C and Z section beams where the compressionflange is unrestrained in the constant moment span over a distanceof 1,626 mm �64 in.� indicate that distortional buckling is themost likely failure mode. Distortional failures occur even whenlocal buckling is at a lower critical elastic moment than distor-

Fig. 25. Comparison of direct strength method prediction equationswith test results

tional buckling. Laterally braced C and Z sections which are free


to fail in distortional buckling result in an average loss of 17% oftheir bending capacity when compared with the same sectionwhere distortional buckling is restrained and failure occurs due tolocal buckling. Previous testing by Yu and Schafer �2003� dem-onstrated that if additional rotational restraint can be provided tothe compression flange, such as through engagement of athrough-fastened deck, the distortional buckling mode can beavoided and a local buckling mode triggered instead.

Comparison of the experimental results with design specifica-tions indicate that the American Specification �AISI 1996� andCanadian Standard �CSA 1994� as well as the newly adoptedNorth American Specification �AISI 2001� provide a poor predic-tion of the strength for members with failures initiated by distor-tional buckling. Errors are, on average, 8–15% unconservative forthese design specifications. Eurocode �CEN 2002� which providesadditional measures to account for distortional buckling is, onaverage, 4% unconservative. Two methods which include explicitprocedures for distortional buckling, the Australian/New Zealandstandard �AS/NZS 1996� and the Direct Strength Method �AISI2004, Appendix 1�, provide reliable predictions of the capacity indistortional buckling with conservative errors of, on average, 2%.The test data provides validation for the Direct Strength Methodpredictor equations, and these equations are recommended for usein design.

Acknowledgments

The sponsorship of the American Iron and Steel Institute �AISI�for both phases of this work, and the Metal Building Manufactur-ers Association for the first phase is gratefully acknowledged.Donation of materials by Varco-Pruden, Clark Steel, and DietrichDesign Group is also gratefully acknowledged. The assistance ofthe AISI task group in developing the testing plan is appreciated.Don Johnson, Maury Golovin, Joe Nunnery, Joe Wellinghoff, andSteve Thomas have all been helpful with their ideas and generouswith their time. Special thanks go to Sandor Ádány from Budap-est University of Technology and Economics, who helped on theEurocode calculation. The experiments would not have been pos-sible without the help of Johns Hopkins undergraduates: LiakosAriston, Brent Bass, Andrew Myers, Sam Phillips, and Tim Ruth.Sam’s work during Phase 1 and Tim’s during Phase 2 was par-ticularly helpful. Finally, the work has been skillfully aided byour laboratory technician Jack Spangler, and our machinist Jim

Table 6. Summary of Test-to-Predicted Ratios for Existing and Proposed

Mtest /MAISI

Local buckling tests Controlling specimens � 1.01

0.04

Second specimens � 1.00

0.05

Distortional buckling tests Controlling specimens � 0.86

0.08

Second specimens � 0.85

0.07

Note: ��average; and �standard deviation.

Kelly.

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Notation

The following symbols are used in this paper:

b � flange width;bc � out-to-out compression flange width;bt � out-to-out tension flange width;

b1 ,b2 � effective width of the compressive portions ofthe web;

bcomp � depth of the full compression portion of theweb;

d � flange lip width;dc � out-to-out compression flange lip stiffener

length;dt � out-to-out compression flange lip stiffener

length;E � modulus of elasticity;

fcr � critical buckling stress;fu � ultimate stress capacity;fy � yield stress;h � out-to-out web depth;

MAISI � AISI �1996� predicted flexural capacity;MAS/NZS � AS/NZS �1996� predicted flexural capacity;

Mcrd � elastic critical distortional buckling moment;Mcr� � elastic critical local buckling moment;

McrLTB � elastic critical lateral-torsional bucklingmoment;

MDSd � DSM predicted flexural capacity fordistortional buckling;

MDS� � DSM predicted flexural capacity for localbuckling;

MDSM � minimum value of MDSd and MDSl;MEN1993 � EN1993 �CEN 2002� predicted flexural

capacity;MNAS � NAS �AISI 2001� predicted flexural capacity;MS136 � S136 �CSA 1994� predicted flexural capacity;Mtest � tested flexural capacity;

My � yield moment;PcrD � actuator load leads to critical elastic

distortional buckling;PcrL � actuator load leads to critical elastic

distortional buckling;PD � actuator load leads to distortional buckling

capacity;PL � actuator load leads to local buckling capacity;Py � actuator load leads to yield meoment;rdc � outer radius between compression flange and

lip;

n Methods

t /MS136 Mtest /MNAS Mtest /MAS/NZS Mtest /MEN1993 Mtest /MDSM

.06 1.02 1.01 1.01 1.03

.04 0.05 0.04 0.06 0.06

.05 1.01 1.00 1.01 1.04

.06 0.07 0.05 0.06 0.07

.92 0.88 1.02 0.96 1.02

.08 0.09 0.07 0.09 0.07

.90 0.87 1.00 0.94 1.00

.07 0.09 0.07 0.09 0.07

Desig

Mtes

1

0

1

0

0

0

0

0



rdt � outer radius between tension flange and lip;rhc � outer radius between web and compression

flange;rht � outer radius between web and tension flange;

t � base metal thickness;�c � compression flange stiffener angle from

horizontal; and�t � tension flange stiffener angle from horizontal.

References

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